Here's two examples:
I'm testing the effectiveness of a drug. Drugs of this class have a certain likelihood of working, the noise in my data is known, the experimental group did this much better than the control... does the drug really work? So far so trivial, in either Bayesianism or Frequentism. Now, I happen to mention that I tested 10000 variants of this drug and only sent data for the one that seemed to work. The rest aren't interesting after all. Under Frequentism, it's easy to take this into account. Under Bayesianism, it requires complex definitions of observations, and is easy to overlook as there's no space for it in the formula.
I have a collection of unfair dice. Unfortunately, they all look the same and got dumped on the floor. Now someone grabbed one off the floor at random and wants to make bets with me about it. Even experienced Bayesians are likely to mix up their propositions in a case like this. I say that from having read discussions of similar problems. Yes, if you do it right, it comes out correctly, but Frequentism makes sure you've thought about what you're asking in the same way Bayesianism makes sure you've thought about your priors.
Somebody else will have to give a third example.
Bayesianism and Frequentism are based on the same math, and math is math. If you use them correctly, they'll get you the same answer every time. The difference is what they make easy, and what mistakes they protect you against.